Graph (x^2)/36-(y^2)/9=1

Problem

(x2)/36−(y2)/9=1

Solution

1. Identify the type of conic section. Since the equation is in the form (x2)/(a2)−(y2)/(b2)=1 it represents a horizontal hyperbola centered at the origin (0,0)

2. Determine the values of a and b We have a2=36 so a=6 We have b2=9 so b=3

3. Locate the vertices. For a horizontal hyperbola, the vertices are at (±a,0) which are (6,0) and (−6,0)

4. Find the equations of the asymptotes. The asymptotes for a horizontal hyperbola are given by y=±b/a*x

5. Substitute the values to find the specific asymptotes.

y=±3/6*x

y=±1/2*x

6. Calculate the foci using the relation c2=a2+b2

c2=36+9

c2=45

c=√(,45)=3√(,5)

The foci are located at (±3√(,5),0)

7. Sketch the graph by plotting the vertices, drawing the asymptotes, and drawing the two branches of the hyperbola opening to the left and right.

Final Answer

(x2)/36−(y2)/9=1* is a horizontal hyperbola with vertices *(±6,0)* and asymptotes *y=±1/2*x

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